## The Annals of Applied Probability

### Limit theory for point processes in manifolds

#### Abstract

Let $Y_{i}$, $i\geq1$, be i.i.d. random variables having values in an $m$-dimensional manifold $\mathcal{M}\subset\mathbb{R}^{d}$ and consider sums $\sum_{i=1}^{n}\xi(n^{1/m}Y_{i},\{n^{1/m}Y_{j}\}_{j=1}^{n})$, where $\xi$ is a real valued function defined on pairs $(y,\mathcal{Y} )$, with $y\in\mathbb{R}^{d}$ and $\mathcal{Y}\subset\mathbb{R}^{d}$ locally finite. Subject to $\xi$ satisfying a weak spatial dependence and continuity condition, we show that such sums satisfy weak laws of large numbers, variance asymptotics and central limit theorems. We show that the limit behavior is controlled by the value of $\xi$ on homogeneous Poisson point processes on $m$-dimensional hyperplanes tangent to $\mathcal{M}$. We apply the general results to establish the limit theory of dimension and volume content estimators, Rényi and Shannon entropy estimators and clique counts in the Vietoris–Rips complex on $\{Y_{i}\}_{i=1}^{n}$.

#### Article information

Source
Ann. Appl. Probab., Volume 23, Number 6 (2013), 2161-2211.

Dates
First available in Project Euclid: 22 October 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1382447685

Digital Object Identifier
doi:10.1214/12-AAP897

Mathematical Reviews number (MathSciNet)
MR3127932

Zentralblatt MATH identifier
1285.60021

#### Citation

Penrose, Mathew D.; Yukich, J. E. Limit theory for point processes in manifolds. Ann. Appl. Probab. 23 (2013), no. 6, 2161--2211. doi:10.1214/12-AAP897. https://projecteuclid.org/euclid.aoap/1382447685

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